In this paper, we consider Markov Decision Processes (MDPs) with error states. Error states are those states entering which is undesirable or dangerous. We define the risk with respect to a policy as the probability of entering such a state when the policy is pursued. We consider the problem of finding good policies whose risk is smaller than some user-specified threshold, and formalize it as a constrained MDP with two criteria. The first criterion corresponds to the value function originally given. We will show that the risk can be formulated as a second criterion function based on a cumulative return, whose definition is independent of the original value function. We present a model free, heuristic reinforcement learning algorithm that aims at finding good deterministic policies. It is based on weighting the original value function and the risk. The weight parameter is adapted in order to find a feasible solution for the constrained problem that has a good performance with respect to the value function. The algorithm was successfully applied to the control of a feed tank with stochastic inflows that lies upstream of a distillation column. This control task was originally formulated as an optimal control problem with chance constraints, and it was solved under certain assumptions on the model to obtain an optimal solution. The power of our learning algorithm is that it can be used even when some of these restrictive assumptions are relaxed.

No abstract

No abstract

No abstract

Classical learning algorithms from the fields of artificial neural networks and machine learning, typically, do not take any costs into account or allow only costs depending on the classes of the examples that are used for learning. As an extension of class dependent costs, we consider costs that are example, i.e. feature and class dependent. We present a natural cost-sensitive extension of the support vector machine (SVM) and discuss its relation to the Bayes rule. We also derive an approach for including example dependent costs into an arbitrary cost-insensitive learning algorithm by sampling according to modified probability distributions.